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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscrngo | Structured version Visualization version GIF version |
Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
Ref | Expression |
---|---|
iscrngo | ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-crngo 35266 | . 2 ⊢ CRingOps = (RingOps ∩ Com2) | |
2 | 1 | elin2 4173 | 1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 RingOpscrngo 35166 Com2ccm2 35261 CRingOpsccring 35265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-crngo 35266 |
This theorem is referenced by: iscrngo2 35269 iscringd 35270 crngorngo 35272 fldcrng 35276 isfld2 35277 isdmn2 35327 |
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